# Density of pythagorean triplets – Part 2

Several people really liked my previous puzzle, including some high-school students 🙂 Some of you have sent me photos of your hand-written proofs. This level of enthusiasm for mathematics is totally awesome. Here is a detailed proof with some beautiful diagrams by Chris Grossack.

Here is my next puzzle:

Let $x_1 < x_2$ and $a^2 + {x_1}^2 = {y_1}^2$ and $a^2 + {x_2}^2 = {y_2}^2$.

Let $x_3 < x_4$ and $b^2 + {x_3}^2 = {y_3}^2$ and $b^2 + {x_4}^2 = {y_4}^2$.

Also, $x_2 - x_1 = x_4 - x_3 = \delta$

All the above variables are positive integers and $a \neq b$ and $x_1 \neq x_3$ and $x_2 \neq x_4$

Prove or disprove the following claim:

Claim: There exists a rational number $0 < q < \delta$ such that $\sqrt{a^2 + {(x_1 + q)}^2}$ and $\sqrt{b^2 + {(x_3 + q)}^2}$ are rational numbers.

If you want to take small baby steps towards a proof, start with the following special cases:

Special case 1: $x_1 = x_3$ and $x_2 = x_4$

Special case 2: $x_1 = x_3$ and $x_2 = x_4$ and $a = b$

Quick homework problem: Prove the Special case 2 using a proof of my previous puzzle.

Have fun solving.

Some mathematician has said pleasure lies not in discovering truth, but in seeking it 🙂

# Density of pythagorean triplets

I designed this simple math puzzle last week:

Prove or disprove the following statement:

• Claim: Let $p, q, r, s$ be positive integers such that $p < q$ and $r < s$. Given two rational numbers $\frac{p}{q} < \frac{r}{s}$, there exist positive integers $a, b$ such that $a < b$ and $\frac{p}{q} < \frac{a}{b} < \frac{r}{s}$ and $a^2 + b^2$ is a perfect square.
• If the above statement is true, show an example of $a, b$, expressed in terms of $p, q, r, s$.
• If the above statement is false, construct a counterexample.